CORRELATION AND REGRESSION
CORRELATION AND REGRESSION
Visualization of Linear Models
Correlation and Regression
Possums > ggplot(data = possum, aes(y = totalL, x = tailL)) + geom_point()
Correlation and Regression
Through the origin > ggplot(data = possum, aes(y = totalL, x = tailL)) + geom_point() + geom_abline(intercept = 0, slope = 2.5)
Correlation and Regression
Through the origin, be!er fit > ggplot(data = possum, aes(y = totalL, x = tailL)) + geom_point() + geom_abline(intercept = 0, slope = 1.7)
Correlation and Regression
Not through the origin > ggplot(data = possum, aes(y = totalL, x = tailL)) + geom_point() + geom_abline(intercept = 40, slope = 1.3)
Correlation and Regression
The "best" fit line > ggplot(data = possum, aes(y = totalL, x = tailL)) + geom_point() + geom_smooth(method = "lm")
Correlation and Regression
Ignore standard errors > ggplot(data = possum, aes(y = totalL, x = tailL)) + geom_point() + geom_smooth(method = "lm", se = FALSE)
CORRELATION AND REGRESSION
Let’s practice!
CORRELATION AND REGRESSION
Understanding the linear model
Correlation and Regression
Generic statistical model response = f(explanatory) + noise
Correlation and Regression
Generic linear model
response = intercept + (slope * explanatory) + noise
Correlation and Regression
Regression model
Correlation and Regression
Fi!ed values
Correlation and Regression
Residuals
Correlation and Regression
Fi!ing procedure
Correlation and Regression
Least squares ●
Easy, deterministic, unique solution
●
Residuals sum to zero
●
Line must pass through
●
Other criteria exist—just not in this course
Correlation and Regression
Key concepts ●
Y-hat is expected value given corresponding X
●
Beta-hats are estimates of true, unknown betas
●
Residuals (e's) are estimates of true, unknown epsilons
●
"Error" may be misleading term—be!er: noise
CORRELATION AND REGRESSION
Let’s practice!
CORRELATION AND REGRESSION
Regression vs. regression to the mean
Correlation and Regression
Heredity ●
Galton's "regression to the mean"
●
Thought experiment: consider the heights of the children of NBA players
Correlation and Regression
Galton's data
Correlation and Regression
Regression modeling ●
"Regression": techniques for modeling a quantitative response
●
Types of regression models: ●
Least squares
●
Weighted
●
Generalized
●
Nonparametric
●
Ridge
●
Bayesian
●
…
CORRELATION AND REGRESSION
Let’s practice!
Visualization of Linear Models
Correlation and Regression
Possums > ggplot(data = possum, aes(y = totalL, x = tailL)) + geom_point()
Correlation and Regression
Through the origin > ggplot(data = possum, aes(y = totalL, x = tailL)) + geom_point() + geom_abline(intercept = 0, slope = 2.5)
Correlation and Regression
Through the origin, be!er fit > ggplot(data = possum, aes(y = totalL, x = tailL)) + geom_point() + geom_abline(intercept = 0, slope = 1.7)
Correlation and Regression
Not through the origin > ggplot(data = possum, aes(y = totalL, x = tailL)) + geom_point() + geom_abline(intercept = 40, slope = 1.3)
Correlation and Regression
The "best" fit line > ggplot(data = possum, aes(y = totalL, x = tailL)) + geom_point() + geom_smooth(method = "lm")
Correlation and Regression
Ignore standard errors > ggplot(data = possum, aes(y = totalL, x = tailL)) + geom_point() + geom_smooth(method = "lm", se = FALSE)
CORRELATION AND REGRESSION
Let’s practice!
CORRELATION AND REGRESSION
Understanding the linear model
Correlation and Regression
Generic statistical model response = f(explanatory) + noise
Correlation and Regression
Generic linear model
response = intercept + (slope * explanatory) + noise
Correlation and Regression
Regression model
Correlation and Regression
Fi!ed values
Correlation and Regression
Residuals
Correlation and Regression
Fi!ing procedure
Correlation and Regression
Least squares ●
Easy, deterministic, unique solution
●
Residuals sum to zero
●
Line must pass through
●
Other criteria exist—just not in this course
Correlation and Regression
Key concepts ●
Y-hat is expected value given corresponding X
●
Beta-hats are estimates of true, unknown betas
●
Residuals (e's) are estimates of true, unknown epsilons
●
"Error" may be misleading term—be!er: noise
CORRELATION AND REGRESSION
Let’s practice!
CORRELATION AND REGRESSION
Regression vs. regression to the mean
Correlation and Regression
Heredity ●
Galton's "regression to the mean"
●
Thought experiment: consider the heights of the children of NBA players
Correlation and Regression
Galton's data
Correlation and Regression
Regression modeling ●
"Regression": techniques for modeling a quantitative response
●
Types of regression models: ●
Least squares
●
Weighted
●
Generalized
●
Nonparametric
●
Ridge
●
Bayesian
●
…
CORRELATION AND REGRESSION
Let’s practice!
